STA261H1 Lecture Notes - Lecture 1: Moment-Generating Function, Generating Function, Random Variable
Document Summary
If v(x) is a montone function of x over the range of x then (cid:12)(cid:12)(cid:12)(cid:12) d dy (cid:12)(cid:12)(cid:12)(cid:12) fx (v 1(y )) fy (y ) = v 1(y ) The moment generating function (mgf) for x is denoted mx (t) and given by etk p(x = k) if x is discrete (cid:104) etx(cid:105) Mx (t) = e (cid:88) (cid:90) all k etx fx (x) dx if x is continuous at all values of t for which the expected value exists. Theorem if a random variable x has an mgf mx (t) then (cid:104) Proof: it is simple as long as we can interchange the order of summation/integral and di erentiations. Examples: find the mean and variance of a random variable with distribution pois( ) and exp( ) Uniquenesstheorem if two random variables x , y have mgfs. Mx and my respectively then if both moments exist and.
